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Quasithin Group
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. The classification of quasithin groups is a crucial part of the classification of finite simple groups. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of ''G''. When ''G'' is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group). Classification The quasithin groups were classified in a 1221-page paper by . An earlier announcement by of the classification, on the basis of which the classification of finite simple groups was announced as finished in 1983, was pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with Index of a subgroup, index 2 and has therefore factorial, ''n''!/2 elements. It is the kernel (algebra), kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian group, abelian if and only if and simple group, simple if and only if or . A5 is the smallest non-abelian simple group, having order of a group, order 60, and thus the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J1
In the area of modern algebra known as group theory, the Janko group ''J1'' is a sporadic simple group of Order (group theory), order : 175,560 = 233571119 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was originally described by Zvonimir Janko in 1965. It is the only Janko group whose existence was proved by Janko himself and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century. Its discovery launched the modern theory of sporadic groups. In 1986 Robert Arnott Wilson, Robert A. Wilson showed that ''J1'' cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the pariah group, pariahs. Properties The smallest faithful complex representation of ''J1'' has dimension 56. ''J1'' can be characterized abstractly as the unique simple group with abelian Sylow theorems, 2-Sylow subgroups and with an Involution (mathematics), involution whose centralizer is isomorphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Solomon
Ronald "Ron" Mark Solomon (b. 15 December 1948 ) is an American mathematician specializing in the theory of finite groups. Solomon studied as an undergraduate at Queens College, City University of New York, Queens College and received a PhD in 1971 at Yale University under Walter Feit with a thesis entitled ''Finite Groups with Sylow 2-Subgroups of the Type of the Alternating Group on Twelve Letters''. In 1972, he began his participation in the Classification of finite simple groups, classification program for finite simple groups, after hearing a lecture by Daniel Gorenstein. He was for two years an instructor at the University of Chicago and the academic year 1974–1975 at Rutgers University, before he became a professor at Ohio State University, where he has remained. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Lyons (mathematician)
Richard Neil Lyons (born January 22, 1945, in New York City, New York) is an American mathematician, specializing in finite group theory. Lyons received his PhD in 1970 at the University of Chicago under John Griggs Thompson with a thesis entitled ''Characterizations of Some Finite Simple Groups with Small 2-Rank''. From 1972 to 2017, he was a professor at Rutgers University. With Daniel Gorenstein and Ronald Solomon he wrote, and is continuing to write, now with Inna Capdeboscq, a series on the second-generation proof of the classification program for finite simple groups. Ten volumes of this series have been published so far. He discovered a sporadic group which Charles Sims constructed and called the Lyons group ''Ly''. In 2012, he shared the Leroy P. Steele Prize for Mathematical Exposition, awarded by the American Mathematical Society, with Michael Aschbacher, Stephen D. Smith, and Ronald Solomon. In 2013, he became a fellow of the American Mathematical Society ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Gorenstein
Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician best remembered for his contribution to the classification of finite simple groups. Gorenstein mastered calculus at age 12 and subsequently matriculated at Harvard University, where he earned his bachelor's and master's degrees. During the Second World War, he taught mathematics to military personnel. After the war, he stayed at Harvard and earned his PhD 1950 under the supervision of Oscar Zariski. In his dissertation, Gorenstein introduced a duality principle for plane curves that motivated Alexander Grothendieck's introduction of Gorenstein rings. Gorenstein held posts at Clark University and Northeastern University, before moving to Rutgers University in 1969, where he remained for the rest of his life. He became the founding director of the Center for Discrete Mathematics and Theoretical Computer Science ( DIMACS) at Rutgers in 1989, and remained at this post until his death. He was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudvalis Group
In the area of modern algebra known as group theory, the Rudvalis group ''Ru'' is a sporadic simple group of order : 145,926,144,000 = 214335371329 : ≈ 1. History ''Ru'' is one of the 26 sporadic groups and was found by and constructed by . Its Schur multiplier has order 2, and its outer automorphism group is trivial. In 1982 Robert Griess showed that ''Ru'' cannot be a subquotient of the monster group.Griess (1982) Thus it is one of the 6 sporadic groups called the pariahs. Properties The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer being the Ree group 2''F''4(2), the automorphism group of the Tits group. This representation implies a strongly regular graph srg(4060, 2304, 1328, 1280). That is, each vertex has 2304 neighbors and 1755 non-neighbors, any two adjacent vertices have 1328 common neighbors, while any two non-adjacent ones have 1280 . Its double cover acts on a 28-dimensional lattice over t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Held Group
In the area of modern algebra known as group theory, the Held group ''He'' is a sporadic simple group of order : 4,030,387,200 = 21033527317 : ≈ 4. History ''He'' is one of the 26 sporadic groups and was found by during an investigation of simple groups containing an involution whose centralizer is an extension of the extra special group 21+6 by the linear group L3(2), which is the same involution centralizer as the Mathieu group M24. A second such group is the linear group L5(2). The Held group is the third possibility, and its construction was completed by John McKay and Graham Higman. In all of these groups, the extension splits. The outer automorphism group has order 2 and the Schur multiplier is trivial. Representations The smallest faithful complex representation has dimension 51; there are two such representations that are duals of each other. It centralizes an element of order 7 in the Monster group. As a result the prime 7 plays a spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group
In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups '' J1'', '' J2'', '' J3'' and '' J4'' introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical. History Janko constructed the first of these groups, ''J''1, in 1965 and predicted the existence of ''J''2 and ''J''3. In 1976, he suggested the existence of ''J''4. Later, ''J''2, ''J''3 and ''J''4 were all shown to exist. ''J''1 was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, ''M''11 and ''M''12 having been found in 1861, and ''M''22, ''M''23 and ''M''24 in 1873. The discovery of ''J''1 caused a great "sensation" and "surprise"The group theorist Bertram Huppert said of ''J''1: "There were a very few things that surprised me in my life... There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Groups
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered. Sometimes the notation ''M''8, ''M''9, ''M''10, ''M''20, and ''M''21 is used for related groups (which act on sets of 8, 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid ''M''13 acting on 13 points. ''M''21 is simple, but is not a sporadic group, being isomorphic to the projective special linear group PSL(3,4). History introduced the group ''M''12 as part of an investigation of multiply transitive permuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
